1. **State the problem:** We have the equation describing the total distance traveled by a car: $$55x + 35y = 200$$ where $x$ is the time in hours traveling at 55 mph, and $y$ is the time in hours traveling at 35 mph. We need to find $x$ or $y$ given certain values. 2. **Formula and rules:** The total distance is the sum of distances traveled at each speed: $$\text{distance} = \text{speed} \times \text{time}$$ So, $$55x + 35y = 200$$ 3. **Part a:** If $y = 2.5$ hours, find $x$. Substitute $y=2.5$: $$55x + 35(2.5) = 200$$ Calculate: $$55x + 87.5 = 200$$ Subtract 87.5 from both sides: $$55x = 200 - 87.5$$ $$55x = 112.5$$ Divide both sides by 55: $$x = \frac{112.5}{55}$$ Simplify fraction: $$x = \frac{\cancel{112.5}}{\cancel{55}} = 2.0454545... \approx 2.05 \text{ hours}$$ 4. **Part b:** If $x = 3$ hours, find $y$. Substitute $x=3$: $$55(3) + 35y = 200$$ Calculate: $$165 + 35y = 200$$ Subtract 165 from both sides: $$35y = 200 - 165$$ $$35y = 35$$ Divide both sides by 35: $$y = \frac{35}{35} = 1 \text{ hour}$$ 5. **Part c:** If $y=0$, find total time $x + y$. Substitute $y=0$: $$55x + 35(0) = 200$$ $$55x = 200$$ Divide both sides by 55: $$x = \frac{200}{55} = \frac{\cancel{200}}{\cancel{55}} = 3.6363636... \approx 3.64 \text{ hours}$$ Since $y=0$, total time is $x + y = 3.64 + 0 = 3.64$ hours. **Reasoning:** If the car never slows down to 35 mph, it travels the entire 200 miles at 55 mph, so time is distance divided by speed. **Final answers:** - a) $x \approx 2.05$ hours - b) $y = 1$ hour - c) Total time $= 3.64$ hours